The link that dirty_rotten_squatter posted has some good general suggestions for figuring out where and when to start. I've never thru-hiked it, but from the many thru-hikers that I've met, sometime between March and April seems to be around when the majority of people start their thru-hike down in Georgia. With those people they usually are planning to reach Katahdin around Septemberish. With that said I've also met a few people who've started from Springer in February. You can start any time you damn please, you just need to make sure you're prepared.
macks: I would probably steer clear of Northern New England in mid-October (Maine, New Hampshire, and the northern part of the trail in Vermont). The weather up in the White Mountains is pretty extreme all year round, and is starting to shift towards winter at that point, and I think they close Baxter State Park early to mid October.
With that said, I think you'd be fine hiking pretty much anywhere south of those states. The days are cool, dry and comfortable, the nights are cool, and the leaves are turning (at least in the Mass/CT area). Bugs won't be as bad either.
I guess take all this with a grain of salt, I've never thru-hiked, but I'm basing all of this off of people I've met who have/are in the process. Just check out the site that dirty_rotten_squatter posted, appalachiantrail.org for info. You can also check out whiteblaze.net and trailjournals.com. Poke around those sites for a while and you should probably have sufficient info.